Question

Recall that a 6-bit string is a bit strings of length 6, and a bit string of weight 3, say, is one with exactly three 1's. How many 6-bit strings are there? How many 6-bit strings have weight 0? How many 6-bit strings have weight 1? How many 6-bit strings have weight 3? How many 6-bit strings have weight 5? How many 6-bit strings have weight 6? How many 6-bit strings have weight 7?

Answer 1

There are 64 possible 6-bit strings. 6-bit strings with weights of 0, 1, 3, 5, and 6 have 1, 6, 20, 6, and 1 possible strings respectively. No 6-bit strings exist with a weight of 7.

Step-by-step explanation:

When discussing 6-bit strings, we are considering all possible combinations of 0s and 1s in a sequence that is 6 bits long. To calculate the number of 6-bit strings, we consider that each bit can be either 0 or 1, giving us 2 possibilities for each bit position. Since there are 6 positions, we calculate the total number by raising 2 to the power of 6, which is 64.

To find the number of 6-bit strings with a specific weight, we can use combinatorics. The weight of a bit string is defined as the number of 1s in the string. For example, the number of 6-bit strings with a weight of 0 is just one, because the only possibility is 000000.

For a weight of 1, we have 6 possible strings, because there is only one 1 and it can be in any of the six positions. For a weight of 3, we use the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of positions and k is the number of 1s. So, the number of 6-bit strings with weight 3 is 20, calculated as C(6, 3).

Similarly, for weight 5, we again use combinations and find that there are 6 possible strings. For weight 6, there is only one possible string, which is 111111. There are no 6-bit strings with weight 7 because the string is only 6 bits long, making it impossible to have more than 6 bits set to 1.

Answer 2

1.. Total number of 6 bit strings is 64

2. Number of 6-bit strings with weight of 0 is 1

3. Number of 6-bit strings with weight of 1 is 6

4. Number of 6-bit strings with weight of 3 is 20

5. Number of 6-bit strings with weight of 5 is 6

6. Number of 6-bit strings with weight of 6 is 1

7. Number of 6-bit strings with weight of 7 is 0

Step-by-step explanation:

A bit string is a string that contains 0 and 1 only

1. Total number of 6 bit strings is 2^6 = 64

2. Number of 6 bit strings with weight 0 is 1

Explanation

Weight 0 means a string with no occurrence of 1

Here, we are only interested in occurrence and not order of occurrence

We apply combination formula for this

nCr = n!/(n-r)!r!

n = 6 and r = 0 i.e. no occurrence of 1

6C0 = 6!/(6-0)!0!

6C0 = 6!/6!0!

6C0 = 1

Hence, the number of string with weight 0 (i.e. no occurrence of 1 ) is 1

3. Number of string with weight 1 is 6

Explanation

Weight 0 means a string with exactly 1 occurrence of '1'

Here, we are only interested in occurrence and not order of occurrence

We apply combination formula for this

nCr = n!/(n-r)!r!

n = 6 and r = 1

6C1 = 6!/(6-1)!1!

6C1 = 6!/5!1!

6C1 = 6

Hence, the number of string with weight 6

4. Number of string with weight 3 is 20

Explanation

n = 6 and r = 3

6C3 = 6!/(6-3)!3!

6C3 = 6!/3!3!

6C3 = 20

Hence, the number of string with weight 3 is 20

5. Number of string with weight 5 is 6

Explanation

n = 6 and r = 5

6C5 = 6!/(6-5)!5!

6C5 = 6!/1!5!

6C5 = 6

Hence, the number of string with weight 5 is 6

6. Number of string with weight 6 is 1

Explanation

n = 6 and r = 6

6C6 = 6!/(6-6)!6!

6C6 = 6!/0!6!

6C6 = 1

Hence, the number of string with weight 6 is 1

7. Number of string with weight 7 is 0

Weight of 7 means that a string that has 7 occurrence of 1

The total length of a 6 bit is 6

Since 6 is less than 7, there's no way a bit of weight 7 can occur.

So, the right answer for this is 0.